Can math or mathematical physics lead to ‘ideas without impressions’?

I would take the view (espoused by some mathematical realists) that mathematical objects and relationships can actually be sensed, and that the organ we use to sense them is the mind. Therefore I believe that all mathematical ideas come from sense-experience. Some are more or less encountered in everyday life. Most people have sensed a natural number, most people have not sensed a p-adic number. But it all comes from the senses.
Nowadays, new math ideas tend to emerge from preexisting math ideas. Historically, mathematics was mostly concerned with studying concepts that did emerge from intuition. Even if Euclid, for instance, tried to reduce geometry to emerge from axioms, the reason he cared about things like lines and triangles is that they represent shapes that we experience in our real life.

However, within the last centuries, math has been developing towards increasing layers of abstraction. We learn about the mathematical objects being studied currently and once in a while, someone comes up with a new object to study, with the ambition that it will help us answer questions about things we knew before.

Take geometry for instance. Modern geometry is concerned with studying objects, such as manifolds, that are abstraction of what we mean to be a shape. If you consider algebraic geometry (it is a deep abstraction rabbit hole, so I think it makes for a good example here), it starts with observations such as the fact that a circle can be represented by an equation such as x²+y² = 1. Then it leverages algebra to study the geometry of the circle through its equation. But now, you are able to come up with equations that make sense algebraically but that do not really describe anything that is in the realm of our immediate reality.

In fact, since algebra is taking over, you can consider equations with unknowns x and y that are not numbers but other abstract objects that simply behave enough like numbers that it can all still make sense mathematically speaking. And then, someone comes up and invents a new layer of abstraction. We consider objects on which some functions act in a way that locally behaves like shapes stemming from equations. I am referring here to something called schemes. I would already be very bothered if someone asked me to describe what a scheme is by referring only to sensitive experience rather than mathematical abstraction. And then there still exist several layers of abstract concepts built upon schemes.

Of course, I suppose you could argue that mathematical concepts are a part of my personal sensitive experience, and therefore none of this contradicts Hume's point of view. But writing this, I am thinking about putting the question differently:

In order to work with all these very abstract objects, mathematicians need to build intuition, in order to be able to "guess" at how they will (or should) behave and orientate their reasoning. Now, sometimes it feels like we are trying to reduce these very abstract things to more tangible experience. Like trying to think visually about an abstract geometric concept that is now very far remove from any sort of tangible reality. And sometimes, it maybe feels like we are actually trying to create some new sensitive personal experience out of our understanding of these abstract objects. Like I may have some sensitive intuition as to what is some specific abstract object I often think about myself, but I would certainly not be able to describe it to someone else in a truly meaningful way.
I'm of the belief that mathematical truths are true regardless of whether or not they are known: if reality didn't exist, the concept 2+2=4 would still be true. However, I think that math that people discover must be connected to the senses.

So like:

* All of math comes from a respective set of axioms
* You can trace the most abstract parts of math down to the axioms
* The axioms were created with information from the mind
* Everything (temporarily excluding math) you know comes from sensory experiences
* Therefore, all math is influenced by sensory experiences

I also think you would also need some sensory input to know whether you are right or wrong, which is a pretty important part of doing math lol
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